## Mathematics

# Phi and An Elegant Equation

## Which took me 20 years to prove.

Let me clarify. I was on vacation 20 years ago when I discovered an interesting and, in my opinion, elegant mathematical equation involving Phi.

Life intervened and 20 years later, while perusing one of my personal journals in which I keep math and physics thoughts, I stumbled upon the equation.

So I sat down and wrote a proof.

As far as I know it is original. I have researched and not found this published anywhere.

Note: This is not a story about Phi or the Golden Ratio. More on Phi and its relation to the Golden Ratio can be found here.

It is an equation I discovered, and the proof I subsequently wrote. The elegance of this equation has stuck in my mind for 20 years and I’d like to share with you.

**Equation and Proof**

**Computer Program**

If you want to play with it, I have included a small JavaScript computer program. I wrote this in VSCode.

**Conclusion**

As stated at GoldenNumber.Net,

*Phi** ( Φ = 1.618033988749895… ), most often pronounced fi like “fly,” is simply an irrational **number** like pi ( p = 3.14159265358979… ), but one with many unusual mathematical properties. Unlike pi, which is a **transcendental number**, **phi** is the solution to a quadratic equation*

Phi has many interesting properties and this appears to be another. I have tried something similar Things with PI, e and other irrational and transcendental numbers, but have not been successful in deriving any special equations. I find this an elegant little equation and will be experimenting with it further.

Thanks for reading!